Abstract: We present a sample of 33 spectroscopically confirmed z ~ 3.1
Ly$\alpha$-emitting galaxies (LAEs) in the Cosmological Evolution Survey
(COSMOS) field. This paper details the narrow-band survey we conducted to
detect the LAE sample, the optical spectroscopy we performed to confirm the
nature of these LAEs, and a new near-infrared spectroscopic detection of the [O
III] 5007 \AA\ line in one of these LAEs. This detection is in addition to two
[O III] detections in two z ~ 3.1 LAEs we have reported on previously (McLinden
et al 2011). The bulk of the paper then presents detailed constraints on the
physical characteristics of the entire LAE sample from spectral energy
distribution (SED) fitting. These characteristics include mass, age,
star-formation history, dust content, and metallicity. We also detail an
approach to account for nebular emission lines in the SED fitting process -
wherein our models predict the strength of the [O III] line in an LAE spectrum.
We are able to study the success of this prediction because we can compare the
model predictions to our actual near-infrared observations both in galaxies
that have [O III] detections and those that yielded non-detections. We find a
median stellar mass of 6.9 $\times$ 10$^8$ M$_{\odot}$ and a median star
formation rate weighted stellar population age of 4.5 $\times$ 10$^6$ yr. In
addition to SED fitting, we quantify the velocity offset between the [O III]
and Ly$\alpha$ lines in the galaxy with the new [O III] detection, finding that
the Ly$\alpha$ line is shifted 52 km s$^{-1}$ redwards of the [O III] line,
which defines the systemic velocity of the galaxy.